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A278069 a(n) = hypergeometric([n, -n], [], 1).

%I A278069 #31 Nov 19 2024 22:11:50
%S A278069 1,0,3,-32,465,-8544,190435,-4996032,150869313,-5155334720,
%T A278069 196677847971,-8286870547680,382200680031313,-19152276311294112,
%U A278069 1036167879649219395,-60195061159370501504,3737352803142621672705,-246970483156591884266112,17306865588065164490357443
%N A278069 a(n) = hypergeometric([n, -n], [], 1).
%F A278069 a(-n) = a(n).
%F A278069 a(n) = n! [x^n] (2*x*exp(h(x)/2))/(4*x-h(x)) with h(x) = sqrt(4*x+1)-1.
%F A278069 a(n) ~ (-1)^n * 2^(2*n-1/2) * n^n / exp(n+1/2). - _Vaclav Kotesovec_, Nov 10 2016
%F A278069 (-9-3*n)*a(n+1)+(12*n^2+53*n+52)*a(n+2)+(4*n^2+33*n+63)*a(n+3)+(n+4)*a(n+4) = 0. - _Robert Israel_, Nov 10 2016
%F A278069 a(n) = ((2*n-1)*a(n-2)-8*(1+n*(n-2))*a(n-1))/(2*n-3) for n>=2. - _Peter Luschny_, Nov 10 2016
%F A278069 a(n) = n! * [x^n] exp(x)/(1 + x)^n. - _Ilya Gutkovskiy_, Apr 07 2018
%p A278069 a := n -> hypergeom([n,-n], [], 1): seq(simplify(a(n)), n=0..18);
%p A278069 # Alternatively:
%p A278069 a := proc(n) option remember; `if`(n<2,1-n,
%p A278069 ((2*n-1)*a(n-2)-8*(1+n*(n-2))*a(n-1))/(2*n-3)) end:
%p A278069 seq(a(n), n=0..18);
%t A278069 Table[HypergeometricPFQ[{n, -n}, {}, 1], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 10 2016 *)
%t A278069 a={};For[n=0,n<19,n++,AppendTo[a,(-1)^n*Sum[(-1)^(j-n+1-Mod[n,2])*Product[(2*n-k)*k/(n-k+1),{k,j,n}],{j,1,n+1}]]]; a (* _Detlef Meya_, Sep 05 2023 *)
%o A278069 (Sage)
%o A278069 def a():
%o A278069     a, b, c, d, h, e = 1, 0, 1, 8, 8, 0
%o A278069     yield a
%o A278069     while True:
%o A278069         yield b
%o A278069         e = c; c += 2
%o A278069         a, b = b, (c*a - h*b)//e
%o A278069         d += 16; h += d
%o A278069 A278069 = a()
%o A278069 [next(A278069) for _ in range(19)]
%Y A278069 Cf. A278070, A278071.
%K A278069 sign
%O A278069 0,3
%A A278069 _Peter Luschny_, Nov 10 2016